Solve the problem.

Consider triangle $A B C$ .

($c$) Why is the law of cosines a better method in this case?

Solve the problem.

The polar equation

$$r=\frac{a\left(1-e^2\right)}{1+e \cos \theta}$$

can be used to graph the orbits of the planets, where $a$ is the average distance in astronomical units from the sun and $e$ is a constant called the eccentricity. The sun will be located at the pole.

Graph the orbits of the four closest planets on the same polar grid. Choose a viewing window that results in a graph with nearly circular orbits.

Solve the problem.

Consider triangle $A B C$ .

(a) Use the law of sines to find candidates for the value of angle $C$.

Find the angle between the pair of vectors.

$$\langle 2,1\rangle,\langle-3,1\rangle$$

In the parallel electrical circuit shown in the figure, the impedance Z can be calculated using the equation

$$Z=\frac{1}{\frac{1}{Z_1}+\frac{1}{Z_2}},$$

where $Z_1$ and $Z_2$ are the impedances for the branches of the circuit.

If $Z_1=50+25 i$ and $Z_2=60+20 i$, calculate $Z$.

Solve the equation. Leave answers in trigonometric form.

$$x^4+16=0$$

Solve the problem.

A Heron triangle is a triangle having integer sides and area. Show that given side is a Heron triangle.

($c$) $a=7, b=15, c=20

Solve the problem.

A Heron triangle is a triangle having integer sides and area. Show that given side is a Heron triangle.

(b) $a=13, b=14, c=15$

Solve the problem.

A Heron triangle is a triangle having integer sides and area. Show that given side is a Heron triangle.

(a) $a=11, b=13, c=20$

Solve the equation. Leave answers in trigonometric form.

$$x^3+125=0$$

Find the polar coordinates of the points of intersection of the given curves for the specified interval of $\theta$.

$$r=\sin 2 \theta, r=\sqrt{2} \cos \theta ; 0 \leq \theta<\pi$$

Find the dot product for the pair of vectors.

$$2 \mathbf{i}+4 \mathbf{j},-\mathbf{j}$$

The current I in a circuit with voltage E, resistance R, capacitive reactance $X_c$, and inductive reactance $X_L$ is

$$I=\frac{E}{R+\left(X_L-X_c\right) i} .$$

Find $I$ if $E=12\left(\cos 25^{\circ}+i \sin 25^{\circ}\right), R=3, X_L=4$, and $X_c=6$. Give the answer in rectangular form, with real and imaginary parts to the nearest tenth.

The following theorem is also attributed to Ptolemy: In a quadrilateral inscribed in a circle, the product of the diagonals is equal to the sum of the products of the opposite sides. The circle in the figure has diameter $1$. Explain why the lengths of the line segments are as shown, and then apply Ptolemy's theorem to derive the formula for the sine of the sum of two angles.